A306364: Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).

A306364 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence from A306364

\(P_{n}(x) = \left(1 + 2\,n\right)\,P_{n-1}(x) + \left(1 - x - n^{2} + n^{2} x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3								3
2	12	3							15
3	60	45							105
4	360	540	45						945
5	2520	6300	1575						10395
6	20160	75600	37800	1575					135135
7	181440	952560	793800	99225					2027025
8	1814400	12700800	15876000	3969000	99225				34459425
9	19958400	179625600	314344800	130977000	9823275				654729075
10	239500800	2694384000	6286896000	3929310000	589396500	9823275			13749310575
11	3113510400	42810768000	128432304000	112378266000	28094566500	1404728325			316234143225
12	43589145600	719220902400	2697078384000	3146591448000	1179971793000	117997179300	1404728325		7905853580625
13	653837184000	12749825088000	58436698320000	87655047480000	46018899927000	7669816654500	273922023375		213458046676875
14	10461394944000	237996734976000	1308982042368000	2454341329440000	1718038930608000	429509732652000	30679266618000	273922023375	6190283353629375

b-files: A306364 ?

Technical details

Node id: n-3951e6b9a140d595 · revision 0 · JSON
Definition source: oeis_catalog:A306364
Fingerprint: f6-125f57a26ac8836e

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A306364 triangle 30 terms, 10 rows · tabf · nonn · catalog b-file?

Row sums ? A001147

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        3;
  2:       12,         3;
  3:       60,        45;
  4:      360,       540,        45;
  5:     2520,      6300,      1575;
  6:    20160,     75600,     37800,      1575;
  7:   181440,    952560,    793800,     99225;
  8:  1814400,  12700800,  15876000,   3969000,   99225;
  9: 19958400, 179625600, 314344800, 130977000, 9823275;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+1)*P(n-1,x) - (n^2-1-(n^2-1)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(2*n + 1)*P[n - 1, x] - (n^2 - 1 - (n^2 - 1)*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (2*n + 1)*v[i - 1] - (n^2 - 1 - (n^2 - 1)*x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A306364 triangle, offset 2, from P₀

%e Example

Triangle begins:
   2:        1;
   3:        3;
   4:       12,         3;
   5:       60,        45;
   6:      360,       540,        45;
   7:     2520,      6300,      1575;
   8:    20160,     75600,     37800,      1575;
   9:   181440,    952560,    793800,     99225;
  10:  1814400,  12700800,  15876000,   3969000,   99225;
  11: 19958400, 179625600, 314344800, 130977000, 9823275;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-3)*P(n-1,x) - (n^2-4*n+3-(n^2-4*n+3)*x)*P(n-2,x) for n >= 4, with P(2,x) = 1, P(3,x) = 3.

%t Mathematica

P[2, x_] := 1;
P[3, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(2*n - 3)*P[n - 1, x] - (n^2 - 4*n + 3 - (n^2 - 4*n + 3)*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 2, 11}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N - 1)); v[1] = 1; v[2] = 3; for(n = 4, N, my(i = n - 1); v[i] = (2*n - 3)*v[i - 1] - (n^2 - 4*n + 3 - (n^2 - 4*n + 3)*x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};
T(11)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003